Heterotic - type I superstring duality and low-energy effective actions

We compare order $R^4$ terms in the 10-dimensional effective actions of SO(32) heterotic and type I superstrings from the point of view of duality between the two theories. Some of these terms do not receive higher-loop corrections being related by supersymmetry to `anomaly-cancelling' terms which depend on the antisymmetric 2-tensor. At the same time, the consistency of duality relation implies that the `tree-level' $R^4$ super-invariant (the one which has $\zeta(3)$-coefficient in the sphere part of the action) should appear also at higher orders of loop expansion, i.e. should be multiplied by a non-trivial function of the dilaton.